Showing papers on "Matrix (mathematics) published in 1989"

Geophysical data analysis : discrete inverse theory

Abstract: Preface. Introduction. DESCRIBING INVERSE PROBLEMS Formulating Inverse Problems. The Linear Inverse Problem. Examples of Formulating Inverse Problems. Solutions to Inverse Problems. SOME COMMENTS ON PROBABILITY THEORY Noise and Random Variables. Correlated Data. Functions of Random Variables. Gaussian Distributions. Testing the Assumption of Gaussian Statistics Confidence Intervals. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 1:THE LENGTH METHOD The Lengths of Estimates. Measures of Length. Least Squares for a Straight Line. The Least Squares Solution of the Linear Inverse Problem. Some Examples. The Existence of the Least Squares Solution. The Purely Underdetermined Problem. Mixed*b1Determined Problems. Weighted Measures of Length as a Type of A Priori Information. Other Types of A Priori Information. The Variance of the Model Parameter Estimates. Variance and Prediction Error of the Least Squares Solution. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 2: GENERALIZED INVERSES Solutions versus Operators. The Data Resolution Matrix. The Model Resolution Matrix. The Unit Covariance Matrix. Resolution and Covariance of Some Generalized Inverses. Measures of Goodness of Resolution and Covariance. Generalized Inverses with Good Resolution and Covariance. Sidelobes and the Backus-Gilbert Spread Function. The Backus-Gilbert Generalized Inverse for the Underdetermined Problem. Including the Covariance Size. The Trade-off of Resolution and Variance. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 3: MAXIMUM LIKELIHOOD METHODS The Mean of a Group of Measurements. Maximum Likelihood Solution of the Linear Inverse Problem. A Priori Distributions. Maximum Likelihood for an Exact Theory. Inexact Theories. The Simple Gaussian Case with a Linear Theory. The General Linear, Gaussian Case. Equivalence of the Three Viewpoints. The F Test of Error Improvement Significance. Derivation of the Formulas of Section 5.7. NONUNIQUENESS AND LOCALIZED AVERAGES Null Vectors and Nonuniqueness. Null Vectors of a Simple Inverse Problem. Localized Averages of Model Parameters. Relationship to the Resolution Matrix. Averages versus Estimates. Nonunique Averaging Vectors and A Priori Information. APPLICATIONS OF VECTOR SPACES Model and Data Spaces. Householder Transformations. Designing Householder Transformations. Transformations That Do Not Preserve Length. The Solution of the Mixed-Determined Problem. Singular-Value Decomposition and the Natural Generalized Inverse. Derivation of the Singular-Value Decomposition. Simplifying Linear Equality and Inequality Constraints. Inequality Constraints. LINEAR INVERSE PROBLEMS AND NON-GAUSSIAN DISTRIBUTIONS L1 Norms and Exponential Distributions.

Parallel and distributed computation

Abstract: This book focuses on numerical algorithms suited for parallelization for solving systems of equations and optimization problems Emphasis on relaxation methods of the Jacobi and Gauss-Seidel type, and issues of communication and synchronization Topics covered include: Algorithms for systems of linear equations and matrix inversion; Herative methods for nonlinear problems; and Shortest paths and dynamic programming

Updating the inverse of a matrix

Abstract: The Sherman–Morrison–Woodbury formulas relate the inverse of a matrix after a small-rank perturbation to the inverse of the original matrix. The history of these fomulas is presented and various applications to statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations are discussed. The Sherman-Morrison-Woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix. This paper surveys the history of these formulas and we examine some applications where these formulas are helpful

Approximating the permanent

Abstract: A randomised approximation scheme for the permanent of a 0–1s presented. The task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph; the latter is accomplished by simulating a Markov chain whose states are the matchings in the graph. For a wide class of 0–1 matrices the approximation scheme is fully-polynomial, i.e., runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output. This class includes all dense matrices (those that contain sufficiently many 1’s) and almost all sparse matrices in some reasonable probabilistic model for 0–1 matrices of given density.For the approach sketched above to be computationally efficient, the Markov chain must be rapidly mixing: informally, it must converge in a short time to its stationary distribution. A major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely c...

Qualitative Data Analysis and Interpretation

Abstract: Miles and Huberman’s 1984 “sourcebook” presents a wide-ranging, creative set of ideas and strategies for analyzing qualitative evaluation/research data. A major message of this sourcebook is that important themes and especially patterns in a set of qualitative data can be effectively discerned, interpreted and repo rt ed via the use of displays. Matrix displays, in particular, are advocated for higher-order analyses of patterns in a set of descriptive results.

Some numerical experiments with variable-storage quasi-Newton algorithms

Abstract: This paper describes some numerical experiments with variable-storage quasi-Newton methods for the optimization of some large-scale models (coming from fluid mechanics and molecular biology). In addition to assessing these kinds of methods in real-life situations, we compare an algorithm of A. Buckley with a proposal by J. Nocedal. The latter seems generally superior, provided that careful attention is given to some nontrivial implementation aspects, which concern the general question of properly initializing a quasi-Newton matrix. In this context, we find it appropriate to use a diagonal matrix, generated by an update of the identity matrix, so as to fit the Rayleigh ellipsoid of the local Hessian in the direction of the change in the gradient. Also, a variational derivation of some rank one and rank two updates in Hilbert spaces is given.

On the fast matrix multiplication in the boundary element method by panel clustering

Abstract: The boundary element method (BEM) leads to a system of linear equations with a full matrix, while FEM yields sparse matrices. This fact seems to require much computational work for the definition of the matrix, for the solution of the system, and, in particular, for the matrix-vector multiplication, which always occurs as an elementary. In this paper a method for the approximate matrix-vector multiplication is described which requires much less arithmetical work. In addition, the storage requirements are strongly reduced.

On- and off-line identification of linear state-space models

Abstract: A geometrically inspired matrix algorithm is derived for the identification of statespace models for multivariable linear time-invariant systems using (possibly noisy) input-output (I/O) measurements only. As opposed to other (mostly stochastic) identification schemes, no variance-covariance information whatever is involved, and only a limited number of I/O-data are required for the determination of the system matrices. Hence, the algorithm can be best described and understood in the matrix formalism, and consists of the following two steps. First, a state vector sequence is realized as the intersection of the row spaces of two block Hankel matrices, constructed with I/O-data. Then the system matrices are obtained at once from the least-squares solution of a set of linear equations. When dealing with noisy data, this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition. Also, the algorithm is easily applied to slowly time-...

Metal Matrix Composites: Thermomechanical Behavior

Abstract: Metal Matrix Composites (MMCs) are among the strongest candidates for use as structural materials in many high temperature and aerospace applications. This book is fully referenced, with literature coverage to mid-1988, and a set of problems is included with each chapter.

Matrix nearness problems and applications

Abstract: A matrix nearness problem consists of finding, for an arbitrary matrix A, a nearest member of some given class of matrices, where distance is measured in a matrix norm. A survey of nearness problems is given, with particular emphasis on the fundamental properties of symmetry, positive definiteness, orthogonality, normality, rank-deficiency and instability. Theoretical results and computational methods are described. Applications of nearness problems in areas including control theory, numerical analysis and statistics are outlined.

GROUPABIL1TY: an analysis of the properties of binary data matrices for group technology

Abstract: SUMMARY Block-diagonalization of the machine-component incidence matrix is the first step in the implementation of group technology. Even powerful algorithms will fail to achieve this if the matrix itself is not amenable to block-diagonalization. The present work analyses the properties of the matrix and identifies the standard deviation of the pairwise similarities (Jaccard7rpar; of the vectors as the major factor that decides the groupability of the data set. Many data sets ranging from the perfectly groupable to the most ill structured ones are analysed and presented. The groupability curves show the variation of the property against the relevant factors.

Measures of Modal Controllability and Observability for First- and Second-Order Linear Systems

Abstract: For a system described by the triple (A9B9C) where the matrix A has a set of distinct eigenvalues and a wellconditioned modal matrix, we propose measures of modal controllability and observability. The angles between the left eigenvectors of A and the columns of the matrix B are used to propose modal controllability measures and the angles between the rows of the matrix C, and the right eigenvectors of A are used to propose modal observability measures. Gross measures of controllabili ty of a mode from all inputs and its observability in all outputs are also proposed. These measures are related to other measures suggested in the literature. A closed-form relation between the norm of the residue and the proposed measures is given, thus linking the residue to the unobservability or uncontrollability of the mode. We finally show that the proposed measures can be applied directly to second-order models.

Algebraic multilevel preconditioning methods. I

Abstract: A recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed. It is based on a sequence of nested finite element spaces with the usual nodal basis functions. Using a nodeordering corresponding to the nested meshes, the finite element stiffness matrix is recursively split up into two-level block structures and is factored approximately in such a way that any successive Schur complement is replaced (approximated) by a matrix defined recursively and thereform only implicitely given. To solve a system with this matrix we need to perform a fixed number (v) of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level. It is shown that by a proper choice of iteration parameters it suffices to use $$v > \left( {1 - \gamma ^2 } \right)^{ - \tfrac{1}{2}} $$ iterations for the so constructedv-foldV-cycle (wherev=2 corresponds to aW-cycle) preconditioning matrices to be spectrally equivalent to the stiffness matrix. The conditions involve only the constant ? in the strengthened C.-B.-S. inequality for the corresponding two-level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance. If we use successive uniform refinements of the meshes the method is of optimal order of computational complexity, if $$\gamma ^2< \tfrac{8}{9}$$ . Under reasonable assumptions of the finite element mesh, the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method, for instance.

Hyperspherical Harmonics: Applications in Quantum Theory

Abstract: Harmonic polynomials.- Generalized angular momentum.- Gegenbauer polynomials.- Fourier transforms in d dimensions.- Fock's treatment of hydrogenlike atoms and its generalization.- Many-dimensional hydrogenlike wave functions in direct space.- Solutions to the reciprocal-space Schroedinger equation for the many-center Coulomb problem.- Matrix representations of many-particle Hamiltonians in hyper spherical coordinates.- Iteration of integral forms of the Schroedinger equation.- Symmetry-adapted hyperspherical harmonics.- The adiabatic approximation.- Appendix A: Angular integrals in a 6-dimensional space.- Appendix B: Matrix elements of the total orbital angular momentum operator.- Appendix C: Evaluation of the transformation matrix U.- Appendix D: Expansion of a function about another center.- References.

Analysis of the linear programming gradient method for optimal design of water supply networks

Abstract: A theoretical analysis of the linear programming (LP) gradient method for optimal design of water distribution networks is presented. The method was first proposed by A. Alperovits and U. Shamir (1977) and has received much attention in the last 10 years. It consists of two stages that are solved in alteration: (1) a LP problem is solved for a given feasible flow distribution and (2) a search is conducted in the space of flow variables, based on the gradient of the objective function (GOF). In this paper a matrix formulation is given for both stages using well-known graph theory matrices. It is proven that the mathematical expression of the GOF is independent of the choice of the sets of loops and paths along which the head constraints are formulated. This is contrary to the claim made by I. C. Goulter et al. (1986). The original GOF expression is shown to have been an approximation of the steepest direction, but still gives good results. Finally, the search procedure is improved by using the projected gradient method.

A general-purpose version of the fast decoupled load flow

Abstract: A new version of the fast decoupled load flow, in which a more broad range of power systems can be solved, is presented. The key lies in the different way in which the resistances are ignored and in a different iteration scheme. In the standard algorithm the resistances are ignored while building the B' load flow matrix: it is shown that it is preferable that the resistances are ignored in the B" matrix instead of the B' matrix. For normal test systems there is hardly any difference in the number of iterations. However, the new algorithm iterates faster if one or more problematic R/X ratios are present. An iteration scheme with strict successive P and Q iterations prevents cycling convergence behavior which can be found in some low voltage systems. The advantages of the new version are demonstrated with runs on IEEE test systems, with both uniformly and nonuniformly scaled reactances. R-scaling up to 3 is always possible, and sometimes values up to 5 can be used. X-scaling of at least 0.1 is possible without losing convergence and with iteration counts which are significantly lower than with the standard scheme. >

New Theoretical Model of Stress Transfer Between Fibre and Matrix in a Uniaxially Fibre-Reinforced Composite

Abstract: A new analytical method has been developed that can predict the stress transfer between fibre and matrix in a uniaxially fibre-reinforced composite associated with either a single matrix crack or a fibre break. Account is taken of thermal residual stresses arising from a mismatch in thermal expansion coefficients between the fibre and matrix. In addition Poisson ratio mismatches are also taken into account. The theoretical approach retains all relevant stress and displacement components, and satisfies exactly the equilibrium equations, the interface conditions and other boundary conditions involving stresses. Two of the four stress-strain-temperature relations are satisfied exactly, whereas the remaining two are satisfied in an average sense. The required non-interface displacement boundary conditions are also satisfied in an average sense. The general representation is used to solve three types of stress transfer problem. A matrix crack and a broken fibre are analysed for the case when there is perfect bonding between fibre and matrix. The third type of problem takes account of frictional slip at the interface governed by the Coulomb friction law. The approximate analytic approach described in this paper, and the preliminary numerical predictions presented, indicate that the stress transfer between fibres and matrix in a unidirectional fibre-reinforced composite, loaded in tension, can now be investigated theoretically in more detail than before. The paper includes some discussion of singularities in the stress fields, which are smoothed by the averaging techniques employed in the analysis. The analytical approach has enabled the development of a micro-mechanical model of frictional slip at the fibre-matrix interface based on the Coulomb friction law, which is more realistic than assuming that the interfacial shear stress is a constant. Predictions are presented of the stress distributions along the fibre-matrix interface and, in particular, it is shown how the length of the frictional slip zone is related to applied strain, friction coefficient, fibre volume fraction and the difference between the test and ‘manufacturing’ temperatures. An indication is given of many other areas of composite modelling where the new theory will be applied.

Fourier series expansion of the transfer equation in the atmosphere-ocean system

Abstract: We consider radiative transfer in a plane-parallel atmosphere bounded by a rough ocean surface. The problem is solved by using a Fourier series decomposition of the radiation field. For the case of a Lambertian surface as a boundary condition, this decomposition is classically achieved by developing the scattering phase matrix in a series of Legendre functions. For the case of a rough ocean surface, we obtain the decomposition by developing both the Fresnel reflection matrix and the wave facet distribution function in Fourier series. This procedure allows us to derive the radiance field for the case of the ruffled ocean surface, with a computation time only a few percent larger than for the case of a Lambertian surface.

On stabilization and the existence of coprime factorizations

Abstract: Shows that any transfer function matrix whose elements belong to the quotient field of H/sub infinity /, and which is stabilizable, has a matrix fraction representation over H/sub infinity / which is coprime in the sense that a matrix Bezout identity can be satisfied. >

Estimating 2-D angles of arrival via two parallel linear arrays

Abstract: A novel approach for high resolution estimates of two-dimensional (2-D) angles of arrival of multiple narrowband sources for two parallel linear arrays is presented. By fully using the properties of the autocovariance matrices, the authors construct a direction-of-arrival (DOA) matrix. The theoretical analysis shows that the eigenvalues and corresponding eigenvectors of the DOA matrix are related to the 2-D angles of arrival of sources. Thus, they can be used to estimate 2-D angles of arrival separately, and the large computations required for 2-D searches can be avoided. Simulation results are given, showing that the method yields excellent estimates with low variance. >

Worst-case complexity bounds on algorithms for computing the canonical structure of finite Abelian groups and the Hermite and Smith normal forms of an integer matrix

Abstract: An $O(s^5 M(s^2 ))$ algorithm for computing the canonical structure of a finite Abelian group represented by an integer matrix of size s (this is the Smith normal form of the matrix) is presented. Moreover, an $O(s^3 M(s^2 ))$ algorithm for computing the Hermite normal form of an integer matrix of size s is given.The upper bounds derived on the computational complexity of the algorithms above improve the upper bounds given by Kannan and Bachem in [SIAM J. Comput., 8 (1979), pp. 499–507] and Chou and Collins in [SIAM J. Comput., 11 (1982), pp. 687–708].

Block reduction of matrices to condensed forms for eigenvalue computations

Abstract: In this paper we described block algorithms for the reduction of a real symmetric matrix to tridiagonal form and for the reduction of a general real matrix to either bidiagonal or Hessenberg form using Householder transformations. The approach is to aggregate the transformations and to apply them in a blocked fashion, thus achieving algorithms that are rich in matrix-matrix operations. These reductions to condensed form typically comprise a preliminary step in the computation of eigenvalues or singular values. With this in mind, we also demonstrate how the initial reduction to tridiagonal or bidiagonal form may be pipelined with the divide and conquer technique for computing the eigensystem of a symmetric matrix or the singular value decomposition of a general matrix to achieve algorithms which are load balanced and rich in matrix-matrix operations.

Stress transfer in single-fibre composites: effect of adhesion, elastic modulus of fibre and matrix, and polymer chain mobility

Abstract: The stress transfer in single-fibre composites is studied experimentally by determining the critical fibre length to diameter ratio,I c/d, in carbon fibre-epoxy resin or poly (ethylene vinyl acetate) systems. Our results and a great number of others available in the literature are compared with the predictions given, on the one hand, by the analytical approach by Cox and, on the other hand, by the theoretical study using finite element technique by Termonia. First, the influence of the fibre-matrix adhesion is analysed and it is observed, in agreement with Termonia, thatI c/d strongly decreases when the bonding efficiency between the two components is increased. Secondly, assuming a perfect fibre-matrix adhesion, it is shown that the critical fibre aspect ratio is proportional to the square root of the ratio of fibre to matrix elastic modulus, as predicted by Cox. However, two linear relationships are established: the first corresponds to the thermosetting and thermoplastic matrices, while the second corresponds to the elastomeric matrices. The difference between these two kinds of materials is attributed to the great difference in polymer chain mobility as shown by a study of the temperature dependence ofI c/d, particularly in the glass transition temperature zone of the matrices. However, in the case of elastomeric materials, the existence of an interphase layer between the fibre and the matrix, having an elastic modulus close to that of the elastomer in its glassy state, can also explain this particular behaviour.